Initializing A Max Heap 8 4 7 6 9 3 10 1 11 5 2 input array = [-, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

Initializing A Max Heap 1 2 3 4 5 6 7 8 9 7 7 11 10 8 Start at rightmost array position that has a child. Index is n/2.  Heapify

Initializing A Max Heap 1 2 3 4 11 6 7 8 9 10 7 7 5 8 Move to next lower array position.

Initializing A Max Heap 1 2 3 4 11 6 7 8 9 7 7 5 10 8 Heapify!

Initializing A Max Heap 1 2 3 9 11 6 7 8 4 7 7 5 10 8

Initializing A Max Heap 1 2 3 9 11 6 7 8 4 7 7 5 10 8 Move to next lower array position and Heapify.

Initializing A Max Heap 1 2 7 9 11 6 3 8 4 7 7 5 10 8

Initializing A Max Heap 1 2 7 9 11 6 3 8 4 7 7 5 10 8 Move to next lower array position and Heapify.

Initializing A Max Heap 1 11 7 9 6 3 8 4 10 7 7 5 8 Find a home for 2.

Initializing A Max Heap 1 11 7 9 10 6 3 8 4 8 7 7 5 Find a home for 2.

Initializing A Max Heap 1 11 7 9 10 6 3 8 4 7 7 2 8 5 Done, move to next lower array position and Heapify.

Initializing A Max Heap 1 11 7 9 10 6 3 8 4 7 7 2 5 8 Find home for 1.

Initializing A Max Heap 11 7 9 10 6 3 8 4 7 7 2 8 5 Find home for 1.

Initializing A Max Heap 11 10 7 9 6 3 8 4 7 7 2 5 8 Find home for 1.

Initializing A Max Heap 11 10 7 9 5 6 3 8 4 7 7 2 8 Find home for 1.

Initializing A Max Heap 11 10 7 9 5 6 3 8 4 7 7 2 1 8 Done.

Initializing a Max Heap* * Source: CLRS 3rd edition

Time Complexity Height of heap = h. 11 9 7 8 5 6 3 1 4 10 7 7 2 8 Height of heap = h. Number of subtrees with root at level j is <= 2 j-1. Time for each subtree is O(h-j+1).

Complexity Time for level j subtrees is <= 2j-1(h-j+1) = t(j). Total time is t(1) + t(2) + … + t(h-1) = O(n).

Leftist Trees Linked binary tree. Can do everything a heap can do and in the same asymptotic complexity. Can meld two leftist tree priority queues in O(log n) time.

Extended Binary Trees Start with any binary tree and add an external node wherever there is an empty subtree. Result is an extended binary tree.

A Binary Tree

An Extended Binary Tree number of external nodes is n+1

The Function s() For any node x in an extended binary tree, let s(x) be the length of a shortest path from x to an external node in the subtree rooted at x.

s() Values Example

s() Values Example 2 2 1 2 1 1 1 1 1

Properties Of s() If x is an external node, then s(x) = 0. Otherwise, s(x) = min {s(leftChild(x)), s(rightChild(x))} + 1 s() values may be computed easily using a postorder traversal.

Height Biased Leftist Trees A binary tree is a (height biased) leftist tree iff for every internal node x, s(leftChild(x)) >= s(rightChild(x))

A Leftist Tree 2 2 1 2 1 1 1 1 1

Leftist Trees--Property 1 In a leftist tree, the rightmost path is a shortest root to external node path and the length of this path is s(root).

A Leftist Tree 2 2 1 2 1 1 1 1 1 Length of rightmost path is 2.

Leftist Trees—Property 2 The number of internal nodes is at least 2s(root) - 1 Because levels 1 through s(root) have no external nodes. So, s(root) <= log(n+1)

A Leftist Tree Levels 1 and 2 have no external nodes. 2 2 1 2 1 1 1 1 1 1 1 Levels 1 and 2 have no external nodes.

Leftist Trees—Property 3 Length of rightmost path is O(log n), where n is the number of nodes in a leftist tree. Follows from Properties 1 and 2.

Leftist Trees As Priority Queues Min leftist tree … leftist tree that is a min tree. Used as a min priority queue. Max leftist tree … leftist tree that is a max tree. Used as a max priority queue.

A Min Leftist Tree 2 4 3 6 8 5 9 8 6

Some Min Leftist Tree Operations empty() size() top() push() pop() meld() initialize() push() and pop() use meld().

Push Operation push(7) 2 4 3 6 8 5 9 8 6

Push Operation push(7) Create a single node min leftist tree. 2 4 3 6 8 5 9 8 6 Create a single node min leftist tree. 7

Push Operation push(7) Create a single node min leftist tree. 2 4 3 6 8 5 9 8 6 Create a single node min leftist tree. Meld the two min leftist trees. 7

Remove Min (pop) 2 4 3 6 8 5 9 8 6

Remove Min (pop) 2 4 3 6 8 5 9 8 6 Remove the root.

Remove Min (pop) Remove the root. Meld the two subtrees. 2 4 3 6 8 5 9

Meld Two Min Leftist Trees 8 6 9 5 4 3 Traverse only the rightmost paths so as to get logarithmic performance.

Meld Two Min Leftist Trees 4 3 6 8 5 6 9 8 6 Meld right subtree of tree with smaller root and all of other tree.

Meld Two Min Leftist Trees 4 3 6 8 5 6 9 8 6 Meld right subtree of tree with smaller root and all of other tree.

Meld Two Min Leftist Trees 6 4 6 8 8 6 Meld right subtree of tree with smaller root and all of other tree.

Meld Two Min Leftist Trees 6 8 Meld right subtree of tree with smaller root and all of other tree. Right subtree of 6 is empty. So, result of melding right subtree of tree with smaller root and other tree is the other tree.

Meld Two Min Leftist Trees 8 6 Make melded subtree right subtree of smaller root. 6 8 Swap left and right subtree if s(left) < s(right). 6 8

Meld Two Min Leftist Trees 8 6 4 8 6 4 Make melded subtree right subtree of smaller root. Swap left and right subtree if s(left) < s(right).

Meld Two Min Leftist Trees 9 5 3 8 6 4 Make melded subtree right subtree of smaller root. Swap left and right subtree if s(left) < s(right).

Meld Two Min Leftist Trees 3 4 5 6 6 9 8 6 8

Initializing In O(n) Time create n single node min leftist trees and place them in a FIFO queue repeatedly remove two min leftist trees from the FIFO queue, meld them, and put the resulting min leftist tree into the FIFO queue the process terminates when only 1 min leftist tree remains in the FIFO queue analysis is the same as for heap initialization